Two leading collectors have transformed a former dairy in the heart of London into a vast gallery that will compete with the Saatchi collection for the attention of contemporary art lovers.

Frank Cohen, a Mancunian DIY magnate whose collection is second only to Charles Saatchi's, and Nicolai Frahm, a Dane based in London since 1997 with postwar European abstract art among his collecting passions, will stage shows drawing on their respective collections as well as loans from other sources.

Situated near the British Museum, the 12,500 sq ft warehouse was the former milk depot for Express Dairies, and Cohen and Frahm are retaining the building's raw, industrial design. Entry will be free and there will be a bar lounge because, in Cohen's words, "we think that art should be for everyone".

Comparisons with Saatchi, who also opens his gallery to the public, are inevitable. There will be some overlaps, but the Dairy will stage one-man shows rather than the group exhibitions favoured by Saatchi. The opening show in April will be devoted to Swiss artist John Armleder. He is among older artists who, Cohen and Frahm believe, have been eclipsed by the art world's obsession with youth.

"We're trying to give London another space which has a completely different feel [from Saatchi's Chelsea gallery]," said Cohen, 69, who made his fortune with DIY stores, having started as a market trader selling wallpaper from an old ambulance "in the freezing cold seven days a week". The son of a factory machinist, he left school at 15 and recalls that his family seemed never to have money but never starved. Over decades, he has built a collection of modern British artists including LS Lowry, as well as contemporary American, German, Chinese and Indian art.

The Tate showed its appreciation in appointing him a Turner prize judge in 2003, when Grayson Perry won (Perry is in Cohen's collection). He opened a gallery in Wolverhampton, but few people beyond those in the know visited. Cohen closed it, deciding that he would find a space in London.

Talking about the cost of collecting art, Cohen believes there are now so many billionaires worldwide that it is only a matter of time before prices for a single contemporary work soar to $500m and beyond. "There's not a million people out there that can buy these things, but enough to spend $100m and $200m for single works," said Cohen.

Frahm predicted: "Very soon you'll see a $500m-work and, in our lifetime, you'll probably see a $1bn artwork. The reason is that there are [many] new buyers. Everyone who has the money wants to get involved in art. In London, where you used to have no collectors, it's an incredible thing to see. So the prices will keep going up because there are very few great art pieces available."

The world record price was broken last year when Sotheby's sold Edvard Munch's The Scream for $120m (£74m). The buyer remains anonymous, but is assumed to be one of the 1,150 billionaires on the 2012 Forbes list. As the range of expensive items for a super-rich individual to spend on is limited, a unique piece of art must have a special appeal.

Cohen said: "When I was kid growing up, no one made billions. If you had £1m or £2m, you were a big shot – £100m now, you're a poor relation." On a Thai beach this month he overheard two holidaymakers. "They were talking about putting money in the bank. 'I think I'll put £200m in.' This is how people talk. But I don't think they care what they spend. If they want [an artwork], they'll buy it."